Adapted from P. Coles, 1999, *The Routledge Critical
Dictionary of the New Cosmology*, Routledge Inc., New York. Reprinted
with the author's permission. To order this book click here:
http://www.routledge-ny.com/books.cfm?isbn=0415923549

The problem, left unresolved in the standard version
of the **Big Bang theory**, stemming from the impossibility of predicting
*a priori* the value of the **density parameter**
which determines whether
the Universe will expand for ever or will ultimately recollapse. This
shortcoming is ultimately a result of the breakdown of the **laws of
physics** at the initial **singularity** in the Big Bang model.

To understand the nature of the mystery of cosmological flatness, imagine you are in the following situation. You are standing outside a sealed room. The contents are hidden from you, except for a small window covered by a small door. You are told that you can open the door at any time you wish, but only once, and only briefly. You are told that the room is bare, except for a tightrope suspended in the middle about two metres in the air, and a man who, at some indeterminate time in the past, began to walk the tightrope. You know also that if the man falls, he will stay on the floor until you open the door. If he does not fall, he will continue walking the tightrope until you look in.

What do you expect to see when you open the door? One thing is obvious: if the man falls, it will take him a very short time to fall from the rope to the floor. You would be very surprised, therefore, if your peep through the window happened to catch the man in transit from rope to floor. Whether you expect the man to be on the rope depends on information you do not have. If he is a circus artist, he might well be able to walk to and fro along the rope for hours on end without falling. If, on the other hand, he is (like most of us) not a specialist in this area, his time on the rope would be relatively brief. Either way, we would not expect to catch him in mid-air. It is reasonable, on the grounds of what we know about this situation, to expect the man to be either on the rope or on the floor when we look.

This may not seem to have much to do with , but the analogy can be
recognised when we realise that does not have a constant value as
time goes by in the Big Bang theory. In fact, in the standard
**Friedmann models**
evolves in a very peculiar way. At times
arbitrarily close to the Big Bang, these models are all described by a
value of arbitrarily close to
1. To put this another way, consider
the Figure under **density parameter**. Regardless of the behaviour at
later times, all three curves shown get closer and closer near the
beginning, and in particular they approach the **flat universe** line. As
time goes by, models with just
a little greater than 1 in the early
stages develop larger and larger values of , reaching values far
greater than 1 when recollapse begins. Universes that start out with
values of just less than 1
eventually expand much faster than the
flat model, and reach values of very close to 0. In the latter case,
which is probably more relevant given the contemporary estimates of
< 1, the transition from near
1 to a value near 0 is very rapid.

Now we can see the analogy. If
is, say, 0.3. then in the very
early stages of cosmic history it was very close to 1, but less than
this value by a tiny amount. In fact, it really is a tiny amount
indeed! At the **Planck time**, for example, has to differ from 1 only
in the sixtieth decimal place. As time went by, hovered close to the
critical density value for most of the expansion, beginning to diverge
rapidly only in the recent past. In the very near future it will be
extremely close to 0. But now, it is as if we had caught the tightrope
walker right in the middle of his fall. This seems very surprising, to
put it mildly, and is the essence of the flatness problem.

The value of determines the
**curvature of spacetime**. It is helpful
to think about the radius of spatial curvature - the characteristic
scale over which the geometry appears to be non-Euclidean, like the
radius of a balloon or of the Earth. The Earth looks flat if we make
measurements on its surface over distances significantly less than its
radius (about 6400 km). But on scales larger than this the effect of
curvature appears. The curvature radius is inversely proportional to 1
- in such a way that the closer
is to unity, the larger is the
radius. (A flat universe has a radius of infinite curvature.) If is
not too different from 1, the scale of curvature is similar to the
scale of our cosmological **horizon**, something that again appears to be
a coincidence.

There is another way of looking at this problem by focusing on the
Planck time. At this epoch, where our knowledge of the relevant
physical laws is scant, there seems to be only one natural timescale
for evolution, and that is the Planck time itself. Likewise, there is
only one relevant length scale: the **Planck length**. The characteristic
scale of its spatial curvature would have been the Planck length. If
spacetime was not flat, then it should either have recollapsed (if it
were positively curved) or entered a phase of rapid undecelerated
expansion (if it were negatively curved) on a timescale of order the
Planek time. But the Universe has avoided going to either of these
extremes for around 10^{60} Planek times.

These paradoxes are different ways of looking at what has become
known as the cosmological flatness problem (or sometimes, because of
the arguments that are set out in the preceding paragraph, the *age
problem* or the *curvature problem*), and it arises from the
incompleteness of the standard Big Bang theory. That it is such a big
problem has convinced many scientists that it needs a big
solution. The only thing that seemed likely to resolve the conundrum
was that our Universe really is a professional circus artist, to
stretch the above metaphor to breaking point. Obviously,
is not
close to zero, as we have strong evidence of a lower limit to its
value of around 0.1. This rules out the man-on-the-floor
alternative. The argument then goes that must be extremely close to
1, and that something must have happened in primordial times to single
out this value very accurately.

The happening that did this is now believed to be cosmological
inflation, a speculation by Alan **Guth** in 1981 about the very early
stages of the Big Bang model. The **inflationary Universe** involves a
curious change in the properties of matter at very high energies
resulting from a **phase transition** involving a quantum phenomenon
called a **scallar field**. Under certain
conditions, the Universe begins to expand much more rapidly than it
does in standard Friedmann models, which are based on properties of
low-energy matter with which we are more familiar. This extravagant
expansion - the inflation - actually reverses the kind of behaviour
expected for in the standard
models. is driven hard towards
1 when inflation starts, rather than drifting away from it as in the cases
described above.

A clear way of thinking about this is to consider the connection between the value of and the curvature of spacetime. If we take a highly curved balloon and blows it up to an enormous size, say the size of the Earth, then its surface will appear to be flat. In inflationary cosmology, the balloon starts off a tiny fraction of a centimetre across and ends up larger than the entire observable Universe. If the theory of inflation is correct, we should expect to be living in a Universe which is very flat indeed, with an enormous radius of curvature and in which differs from 1 by no more than one part in a hundred thousand.

The reason why cannot be
assigned a value closer to 1 is that
inflation generates a spectrum of **primordial density fluctuations** on
all scales, from the microscopic to the scale of our observable
Universe and beyond. The density fluctuations on the scale of our
horizon correspond to an uncertainty in the mean density of matter,
and hence to an uncertainty in the value of .

One of the problems with inflation as a solution to the flatness
problem is that, despite the evidence for the existence of **dark
matter**, there is no really compelling evidence of enough such material
to make the Universe closed. The question then is that if, as seems
likely, is significantly
smaller than 1, do we have to abandon
inflation? The answer is not necessarily, because some models of
inflation have been constructed that can produce an **open universe**. We
should also remember that inflation predicts a flat universe, and the
flatness could be achieved with a low matter density if there were a
**cosmological constant** or, in the language of particle physics, a
nonzero vacuum energy density.

On the other hand, even if
were to turn out to be very close to 1,
that would not necessarily prove that inflation happened either. Some
other mechanism, perhaps associated with the epoch of **quantum gravity**,
might have trained our Universe to walk the tightrope. It maybe, for
example, that for some reason quantum gravity favours a flat spatial
geometry. Perhaps, then, we should not regard the flatness `problem'
as a problem: the real problem is that we do not have a theory of the
very beginning in the Big Bang cosmology.

FURTHER READING:

Coles, P. and Ellis, G.F.R., *Is the Universe Open or Closed?*
(Cambridge University Press, Cambridge, 1997).
Guth, A.H., `Inflationary Universe: A possible solution to the horizon
and flatness problems', *Physical Review* D, 1981, **23**, 347.
Narlikar, J.V. and Padmanabhan, T., `Inflation for astronomers',
*Annual Reviews of Astronomy and Astrophysics*, 1991, **29**, 325.